refactoring ...

[helm.git] / helm / software / helena / examples / exp_math / L.hln

diff --git a/helm/software/helena/examples/exp_math/L.hln b/helm/software/helena/examples/exp_math/L.hln
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+++ b/helm/software/helena/examples/exp_math/L.hln
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+\require preamble
+
+\* Intuitionistic Predicate Logic with Equality *\
+
+\open elements \* [1] 2.1. 2.2. 3.1 *\
+
+   \decl "logical false" False: *Prop
+
+   \decl "logical conjunction" And: *Prop => *Prop -> *Prop
+
+   \decl "logical disjunction" Or: *Prop => *Prop -> *Prop
+
+\* implication and non-dependent abstraction are isomorphic *\
+   \def "logical implication" 
+      Imp = [p:*Prop, q:*Prop] p -> q : *Prop => *Prop -> *Prop
+
+\* comprehension and dependent abstraction are isomorphic *\
+   \def "unrestricted logical comprehension"
+      All = [q:*Obj->*Prop] [x:*Obj] q(x) : (*Obj -> *Prop) -> *Prop
+
+   \decl "unrestricted logical existence" Ex: (*Obj -> *Prop) -> *Prop
+
+   \decl "syntactical identity" Id: *Obj => *Obj -> *Prop
+
+\close
+
+\open abbreviations \* [1] 2.3. *\
+
+   \def "logical negation" 
+      Not = [p:*Prop] p -> False : *Prop -> *Prop
+
+   \def "logical equivalence"
+      Iff = [p:*Prop, q:*Prop] And(p -> q, q -> p) : *Prop => *Prop -> *Prop
+
+   \def "unrestricted strict logical existence"
+      EX = [p:*Obj->*Prop] Ex([x:*Obj] And(p(x), [y:*Obj] p(y) -> Id(x, y)))
+         : (*Obj -> *Prop) -> *Prop
+
+   \def "negated syntactical identity"
+      NId = [x:*Obj, y:*Obj] Not(Id(x, y)) : *Obj => *Obj -> *Prop
+
+\close